Download 8-1 Simple Trigonometric Equations

Warm Up
• Use your knowledge of
UC to find at least one
value for q.
1) sin 𝜃 =
1
2
2) cos 𝜃 = −
3) tan 𝜃 = 1
3
2
• State as many angles as
you can that are
referenced by each:
1) 30°
2)
𝜋
3
3) 0.65 radians
Useful information to MEMORIZE:
𝜋
3𝜋
≈ 1.57
≈ 4.71
2
2
2𝜋 ≈ 6.28
8-1 Simple Trigonometric
Equations
Objective: To solve simple
Trigonometric Equations and apply
them
There are many solutions to the
1
trigonometric equation sin 𝑥 =
2
y
-19π
6
-3π
-11π
6
-7π
6
π
6
1
5π
6
π
-2π -π
13π
6
17π
6
2π 3π
25π
6
4π
1
y
=
x
2
-1
All the solutions for x can be expressed in the form of
a general solution.
Copyright © by Houghton Mifflin Company, Inc.
All rights reserved.
3
𝜋
6
• We know that 𝑥 = and 𝑥 =
particular solutions.
5𝜋
6
are two
• Since the period of sin 𝑥 is 2𝜋, we can add
integral multiples of 2𝜋 to get the other
solutions:
𝜋
6
• 𝑥 = + 2n𝜋 and 𝑥 =
integer
5𝜋
6
+ 2n𝜋 When 𝑛 is any
Solving for angles that are not on UC
We will work through solutions algebraically and
graphically.
Learning both methods will enhance your
understanding of the work.
y
1
90
180
270
x
360
degrees
-1
Example 1: Find the values of 0° < 𝑥 < 360°for
which sin𝑥 = −0.35
Method 1: Algebraically:
Step 1 Set the calculator in degree mode
and use the inverse sine key
𝑥 ≈ −20.5°
Find the final answer(s) for the given
range.
• Since the answer given by your calculator is
NOT between 0 and 360 degrees, find the
proper answers by using RA.
y
1
90
180
270
x
360
degrees
-1
Solving Graphically
If you had been asked to find ALL values of 𝑥 for
which sin 𝑥 = −0.35 , then your answer would
be:
𝒙 ≈ 𝟐𝟎𝟎. 𝟓 + 𝟑𝟔𝟎𝒏 AND
𝒙 ≈ 𝟑𝟑𝟗. 𝟓 + 𝟑𝟔𝟎𝒏, for any integer 𝑛.
Example 2: Find the values of 𝑥 between
0 and 2𝜋 for which sin 𝑥 = 0.6
Method 1: Algebraically:
Step 1 Set the calculator in radian mode
and use the inverse sine key
Step 2: Determine the proper quadrant
0.6435 is the reference angle for other
solutions.
Since sin 𝑥 is positive, a Quadrant II
angle also satisfies the equation.
𝑥 = 𝜋 − 0.6435 ≈ 2.4981.
Final answers are: 𝒙 ≈ 𝟎. 𝟔𝟒𝟑𝟓 𝒂𝒏𝒅 𝟐. 𝟒𝟗𝟖𝟏.
If you had been asked to find ALL values of 𝑥 for
which sin 𝑥 = 0.6 , then your answer would be:
𝒙 ≈ 𝟎. 𝟔𝟒𝟑𝟓 + 𝟐𝝅𝒏 AND
𝒙 ≈ 𝟐. 𝟒𝟗𝟖𝟏 + 𝟐𝝅𝒏, for any integer 𝑛.
Example 1: Find the values of 𝑥 between
0 and 2𝜋 for which sin 𝑥 = 0.6
Method 2: Graphically:
Step 1 Set the calculator in radian mode.
Use your Knowledge of trig functions
to choose an appropriate window
Use the intersect
Key once more
for the second
point of
intersection. i.e
solution.
When you use the graphing method, you can
easily see there is more than one solution.
When using the graphing method, it might take
a while to set the window properly.
The algebraic method is quicker, however, you
have make sure to look for a possible second
answer.
• To solve an equation involving a single
trigonometric function, we first transform
the equation so that the function is alone
on one side of the equals sign. Then we
follow the same procedure used in
Example 1.
Example 2
To the nearest tenth degree, solve:
𝟑 cos 𝜽 + 𝟗 = 𝟕 for 𝟎𝒐 ≤ 𝜽 ≤ 𝟑𝟔𝟎𝒐
First apply the basic algebra rules and isolate the
variable.
3 cos 𝜃 + 9 = 7
3 cos 𝜃 = −2
cos 𝜃 =
2
−
3
Find the appropriate quadrant
Since cos 𝜃 < 0 , the final answers are in
the QII and QIII.
Use your knowledge of reference angle to
find the second answer:
The final answers are:
𝜃 ≈ 131.8° or 𝜃 ≈ 228.2°
Another way; ignore the negative sign.
The reference angle is:
−1 2
𝑐𝑜𝑠 ( )
3
≈ 48.2𝑜
The first solution is:
𝜃 ≈ 180𝑜 − 48.2𝑜 = 131.8𝑜
The second solution is:
𝜃 ≈ 180𝑜 + 48.2𝑜 = 228.2𝑜
Graphing Calculator:
Although this is a reasonable
window to start with, it does not
capture the graph. So change
Ymin and Ymax.
Warm Up Day 2;
1. 𝑐𝑜𝑠𝜃 = 2
a) has 0 solution.
b) has 1 solution.
c) has 2 solutions.
d) has infinite number of
solutions.
2. 𝑡𝑎𝑛𝜃 = 2
a) has 0 solution.
b) has 1 solution.
c) has 2 solutions.
d) has infinite number of
solutions.
• Graph sine,
cosine and
tangent
functions.
Inclination and Slope
• The inclination of a line is the angle 𝑎 , where
0𝑜 ≤ 𝑎 < 180𝑜 , that is measured from the
positive x-axis to the line.
Inclination and Slope
• The inclination of a line is the angle 𝑎 , where
0𝑜 ≤ 𝑎 < 180𝑜 , that is measured from the
positive x-axis to the line. The line at the left
below has inclination 35𝑜 . The line at the
right below has inclination 155𝑜 . The
theorem that follows states that the slop of a
nonvertical line is the tangent of its
inclination.
Theorem
• For any line with slope 𝑚 and inclination 𝑎
𝑚 = tan 𝑎 if 𝑎 ≠ 90𝑜 .
• If 𝑎 = 90𝑜 , than the line has no slope. (The
line is vertical.)
Example 3
to the nearest degree, find the
inclination of the line 2𝑥 + 5𝑦 = 15
Solution: rewrite the equation as 𝑦 =
2
Slope = - = tan 𝑎
5
2
−1
𝑡𝑎𝑛
− ≈ −21.8𝑜
5
𝑜
𝑎=
angle is 21.8 .)
2
− 𝑥
5
( the reference
+3
• Since tan 𝑎 is negative and 𝑎 is positive angle,
90𝑜 < 𝑎 < 180𝑜 the inclination is 180𝑜 −
21.8𝑜 ≈ 158.2𝑜 .
• In section 6-7, you learned to graph conic
sections whose equations have no 𝑥𝑦 − 𝑡𝑒𝑟𝑚.
That is equation of the form.
𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
• Where B=0. the graph at the right shows conic
section with center at the origin whose
equation has an 𝑥𝑦 − 𝑡𝑒𝑟𝑚 (𝐵 ≠ 0). Conics
like this have one of their two axes inclined at
an angle 𝑎 to the x axis. To find this direction
angle 𝑎 , use the formula below.
𝑎=
𝜋
4
if A=C
Tan 2𝑎 =
𝐵
𝐴−𝐶
if A≠C , and 0< 2𝑎 < 𝜋
The direction angle a is useful in finding the
equation of the axes of these conic sections.
This is shown in method 1 of example 4 on the
next page
Homework:
• Sec 8.1 Written exercises #1-21 odds
• Optional: Sec 8.2 written exercises 22-32 ALL